where the improved stress tensor \(\Theta ^{\mu \nu }\)  is no longer symmetric, but now does obey the conservation equation \(\nabla _\mu \Theta ^{\mu \nu }=0\) . It is clear that our earlier simple-minded derivation of the Weyl anomaly would run into rough waters here. The expression for the Weyl anomaly actually depends on the formulation. For a stress tensor that yields consistency with Wess-Zumino conditions, the Weyl anomaly gets a correction proportional to \(c-\bar{c}\) . For a covariant stress tensor, constructed at the expense of not satisfying the Wess-Zumino conditions, one get the same expression as (REF ), with \(c\)  replaced by \(\frac{1}{2}(c+\bar{c})\) . Finally, for a non-symmetric stress tensor where we have traded the gravitational anomaly for a Lorentz anomaly, the Weyl anomaly is again the same as the covariant case, albeit that the conservation equation now holds. We point the reader to [1]}, [2]} for further details on this.
